Conventional Magnets for Accelerators Lecture 1

Transcription

1 Conventional Magnets for Accelerators Lecture 1 Ben Shepherd Magnetics and Radiation Sources Group ASTeC Daresbury Laboratory ben.shepherd@stfc.ac.uk Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

2 Course Philosophy An overview of magnet technology in particle accelerators, for room temperature, static (dc) electromagnets, and basic concepts on the use of permanent magnets (PMs). Not covered: superconducting magnet technology. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

3 Contents lectures 1 and 2 DC Magnets: design and construction Introduction Nomenclature Dipole, quadrupole and sextupole magnets Higher order magnets Magnetostatics in free space (no ferromagnetic materials or currents) Maxwell's 2 magnetostatic equations Solutions in two dimensions with scalar potential (no currents) Cylindrical harmonic in two dimensions (trigonometric formulation) Field lines and potential for dipole, quadrupole, sextupole Significance of vector potential in 2D Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

4 Contents lectures 1 and 2 Introducing ferromagnetic poles Ideal pole shapes for dipole, quad and sextupole Field harmonics-symmetry constraints and significance 'Forbidden' harmonics resulting from assembly asymmetries The introduction of currents Ampere-turns in dipole, quad and sextupole Coil economic optimisation-capital/running costs Summary of the use of permanent magnets (PMs) Remnant fields and coercivity Behaviour and application of PMs Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

5 Contents lectures 1 and 2 The magnetic circuit Steel requirements: permeability and coercivity Backleg and coil geometry: 'C', 'H' and 'window frame' designs Classical solution to end and side geometries the Rogowsky rolloff Magnet design using FEA software FEA techniques and codes Opera 2D, Opera 3D Judgement of magnet suitability in design Magnet ends computation and design Some examples of magnet engineering Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

6 DC Magnets INTRODUCTION Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

7 Units SI Units Variable Force, F Charge, q Flux density, B (commonly referred to as field ) Magnetic field, H (magnetomotive force produced by electric currents) Current, I Energy, E Unit Newton (N) Coulomb (C) Tesla (T) or Gauss (G) 1 T = 10,000 G Amp/metre (A/m) Ampere (A) Joule (J) Permeability of free space μμ 0 = 4ππ 10 7 T.m/A Charge of 1 electron ee = C 1 ev = 1.6x10-19 J Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

8 Magnetic Fields Flux Density Value Item pt human brain magnetic field 24 µt strength of magnetic tape near tape head µt strength of Earth's magnetic field at 0 latitude (on the equator) 0.5 mt the suggested exposure limit for cardiac pacemakers by American Conference of Governmental Industrial Hygienists (ACGIH) 5 mt the strength of a typical refrigerator magnet 0.15 T the magnetic field strength of a sunspot 1 T to 2.4 T coil gap of a typical loudspeaker magnet 1.25 T strength of a modern neodymium-iron-boron (Nd 2 Fe 14 B) rare earth magnet. 1.5 T to 3 T strength of medical magnetic resonance imaging systems in practice, experimentally up to 8 T 9.4 T modern high resolution research magnetic resonance imaging system 11.7 T field strength of a 500 MHz NMR spectrometer 16 T strength used to levitate a frog 36.2 T strongest continuous magnetic field produced by non-superconductive resistive magnet 45 T strongest continuous magnetic field yet produced in a laboratory (Florida State University's National High Magnetic Field Laboratory in Tallahassee, USA) T strongest (pulsed) magnetic field yet obtained non-destructively in a laboratory (National High Magnetic Field Laboratory, Los Alamos National Laboratory, USA) 730 T strongest pulsed magnetic field yet obtained in a laboratory, destroying the used equipment, but not the laboratory itself (Institute for Solid State Physics, Tokyo) 2.8 kt strongest (pulsed) magnetic field ever obtained (with explosives) in a laboratory (VNIIEF in Sarov, Russia, 1998) 1 to 100 MT strength of a neutron star 0.1 to 100 GT strength of a magnetar Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

9 Why magnets? Analogy with optics Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

10 Magnets are like lenses sort of Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

11 Magnet types Dipoles to bend the beam Quadrupoles to focus it Sextupoles to correct chromaticity Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

12 Magnets - dipoles To bend the beam uniformly, dipoles need to produce a field that is constant across the aperture. But at the ends they can be either: Sector dipole Parallel-ended dipole They have different focusing effect on the beam; (their curved nature is to save material and has no effect on beam focusing). Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

13 Sector dipoles focus horizontally Dipole end focusing The end field in a parallel ended dipole focuses vertically B Off the vertical centre line, the field component normal to the beam direction produces a vertical focusing force. B beam Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

14 Magnets - quadrupoles Quadrupoles produce a linear field variation across the beam. Field is zero at the magnetic centre so that on-axis beam is not bent. Note: beam that is horizontally focused is vertically defocused. These are upright quadrupoles. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

15 Beam that has horizontal displacement (but not vertical) is deflected vertically. Skew Quadrupoles Horizontally centred beam with vertical displacement is deflected horizontally. So skew quadrupoles couple horizontal and vertical transverse oscillations. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

16 Sextupoles In a sextupole, the field varies as the square of the displacement. By 0 0 x Off-momentum particles are incorrectly focused in quadrupoles (e.g., high momentum particles with greater rigidity are underfocused), so transverse oscillation frequencies are modified chromaticity. But off momentum particles circulate with a horizontal displacement (high momentum particles at larger x) So positive sextupole field corrects this effect can reduce chromaticity to 0. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

17 Higher order magnets e.g. Octupoles: Effect? BB yy xx 3 Octupole field induces Landau damping: Introduces tune-spread as a function of oscillation amplitude De-coheres the oscillations Reduces coupling By 0 0 x Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

18 Describing the field MAGNETOSTATICS IN FREE SPACE Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

19 No currents, no steel - Maxwell s static equations in free space In the absence of currents: Then we can put: So that:. BB = 0 HH = jj jj = 0 BB = φφ 22 φφ = 0 (Laplace s equation) Taking the two dimensional case (i.e. constant in the z direction) and solving for polar coordinates (r, θ): φφ = EE + FF θθ GG + HH ln rr + JJ nn rr nn cos nnnn + KK nn rr nn sin nnnn + LL nn rr nn cos nnnn + MM nn rr nn sin nnnn nn=1 Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

20 The scalar potential simplifies to: In practical situations Φ = JJ nn rr nn cos nnnn + KK nn rr nn sin nnnn nn=1 with n integral and J n, K n a function of geometry. Giving components of flux density: BB rr = nnjj nn rr nn 1 cos nnnn + nnkk nn rr nn 1 sin nnnn nn=1 BB θθ = nnjj nn rr nn 1 sin nnnn + nnkk nn rr nn 1 cos nnnn nn=1 Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

21 Physical significance This is an infinite series of cylindrical harmonics; they define the allowed distributions of B in 2 dimensions in the absence of currents within the domain of (r,θ). Distributions not given by above are not physically realisable. Coefficients J n, K n are determined by geometry (remote iron boundaries and current sources). Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

22 In Cartesian Coordinates To obtain these equations in Cartesian coordinates, expand the equations for φ and differentiate to obtain flux densities cos 2θθ = cos 2 θθ sin 2 θθ sin 2θθ = 2 sin θθ cos θθ cos 3θθ = cos 3 θθ 3 cos θθ sin 2 θθ sin 3θθ = 3 sin θθ cos 2 θθ sin 3 θθ cos 4θθ = cos 4 θθ + sin 4 θθ 6 cos 2 θθ sin 2 θθ sin 4θθ = 4 sin θθ cos 3 θθ 4 sin 3 θθ cos θθ etc (messy!); xx = rr cos θθ BB xx = yy = rr sin θθ BB yy = Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

23 n = 1: Dipole field Cylindrical: BB rr = JJ 1 cos θθ + KK 1 sin θθ BB θθ = JJ 1 sin θθ + KK 1 cos θθ φφ = JJ 1 rr cos θθ + KK 1 rr sin θθ Cartesian: BB xx = JJ 1 BB yy = KK 1 φφ = JJ 1 xx + KK 1 yy So, J 1 = 0 gives vertical dipole field: B φ = const. K 1 =0 gives horizontal dipole field. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

24 n = 2: Quadrupole field Cylindrical: BB rr = 2JJ 2 rr cos 2θθ + 2KK 2 rr sin 2θθ BB θθ = 2JJ 2 rr sin 2θθ + 2KK 2 rr cos 2θθ φφ = JJ 2 rr 2 cos 2θθ + KK 2 rr 2 sin 2θθ J 2 = 0 gives 'normal' or right quadrupole field. Cartesian: BB xx = 2(JJ 2 xx + KK 2 yy) BB yy = 2(KK 2 xx JJ 2 yy) φφ = JJ 2 (xx 2 yy 2 ) + 2KK 2 xxxx Line of constant scalar potential K 2 = 0 gives 'skew' quad fields (above rotated by π/4). Lines of flux density Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

25 n = 3: Sextupole field Cylindrical: BB rr = 3JJ 3 rr 2 cos 3θθ + 3KK 3 rr 2 sin 3θθ BB θθ = 3JJ 3 rr 2 sin 3θθ + 3KK 3 rr 2 cos 3θθ φφ = JJ 3 rr 3 cos 3θθ + KK 3 rr 3 sin 3θθ Cartesian: BB xx = 3(JJ 3 (xx 2 yy 2 ) + 2KK 3 xxxx) BB yy = 3(KK 2 (xx 2 yy 2 ) 2JJ 3 xxxx) φφ = JJ 3 (xx 3 3yy 2 xx) + KK 3 (3yyxx 2 yy 3 ) +C -C +C J 3 = 0 giving 'normal' or right sextupole field. -C +C -C Line of constant scalar potential Lines of flux density Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

26 Summary: variation of B y on x axis Dipole - constant field B y B y x Quad - linear variation x B y Sextupole - quadratic variation 0 0 x Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

27 Alternative notation (most lattice codes) Magnet strengths are specified by the value of k n (normalised to the beam rigidity) order n of k is different to the 'standard' notation: BB xx = BBBB nn=0 kknn xx nn nn! k has units: dipole is n = 0 quad is n = 1 etc. k 0 (dipole) m -1 k 1 (quadrupole) m -2 Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

28 Significance of vector potential in 2D We have: and Expanding: = AA zz AA yy ii + BB = AA (A is vector potential). AA = 0 BB = AA AA xx AA zz jj + AA yy AA xx kk where i, j, k are unit vectors in x, y, z. In 2 dimensions BB zz = 0 and / zz = 0 So AA xx = AA yy = 0 and Note: BB = AA zz ii AA zz jj A is in the z direction, normal to the 2D problem.. BB = 2 AA zz xxxxxx 2 AA zz = 0 Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

29 Total flux between two points A In a two-dimensional problem the magnetic flux between two points is proportional to the difference between the vector potentials at those points. B φφ (AA 2 AA 1 ) A 1 A 2 Proof on next slide. Φ Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

30 Proof Consider a rectangular closed path, length λ in z direction at (x 1,y 1 ) and (x 2,y 2 ); apply Stokes theorem: φφ = BB. ddss = AA. ddss = AA. ddss But A is exclusively in the z direction, and is constant in this direction. So: λ A B ds AA. ddss = λλ[aa xx 1, yy 1 AA xx 2, yy 2 ] ds y z (x 1, y 1 ) (x 2, y 2 ) x φφ = λλ[aa xx 1, yy 1 AA xx 2, yy 2 ] Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

31 Going from fields to magnets STEEL POLES AND YOKES Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

32 What is the perfect pole shape? What is the ideal pole shape? Flux is normal to a ferromagnetic surface with infinite µ: µ = µ = 1 curl H = 0 Flux is normal to lines of scalar potential: BB = φ So the lines of scalar potential are the perfect pole shapes! (but these are infinitely long!) therefore H.ds = 0 in steel H = 0 therefore parallel H air = 0 therefore B is normal to surface. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

33 Equations for the ideal pole Equations for Ideal (infinite) poles; (JJ nn = 0) for normal (ie not skew) fields: Dipole: yy = ± gg 2 (g is interpole gap) Quadrupole: xxxx = ± RR2 2 Sextupole: 3xx 2 yy yy 3 = ±RR 3 R Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

34 Combined function (CF) magnets 'Combined Function Magnets' - often dipole and quadrupole field combined (but see next-but-one slide): A quadrupole magnet with physical centre shifted from magnetic centre. Characterised by 'field index' n, positive or negative depending on direction of gradient - do not confuse with harmonic n! B nn = ρρ BB 0 ρ is radius of curvature of the beam B o is central dipole field Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

35 Combined Function Magnets CERN Proton Synchrotron Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

36 Combined function geometry Combined function (dipole & quadrupole) magnet: beam is at physical centre flux density at beam = B 0 gradient at beam = magnetic centre is at B and X = 0 δδbb δδxx separation magnetic to physical centre = X 0 X 0 x magnetic centre, X = 0 physical centre x = 0 Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

37 Pole of a CF dipole & quad magnet Ref geometry as on previous slide: Flux density at beam centre: Gradient across beam: B 0 [T] g [T/m] Displacement of beam from quadrupole centre: X 0 Local displacement from beam centre: X; So: BB 0 = gg XX 0 XX 0 = BB 0 gg And XX = xx + XX 0 Quadrupole equation: So pole equation ref beam centre: Adjust R to satisfy beam dimensions. XXXX = RR2 2 yy = RR2 2 1 xx+xx 0 Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

38 Other combinations: Other combined function magnets dipole, quadrupole and sextupole dipole & sextupole (for chromaticity control) dipole, skew quad, sextupole, octupole (at DL) Generated by pole shapes given by sum of correct scalar potentials amplitudes built into pole geometry not variable multiple coils mounted on the yoke amplitudes independently varied by coil currents Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

39 The SRS multipole magnet Could develop: vertical dipole horizontal dipole upright quad skew quad sextupole octupole others Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

40 The Practical Pole Practically, poles are finite, introducing errors; these appear as higher harmonics which degrade the field distribution. However, the iron geometries have certain symmetries that restrict the nature of these errors. Dipole: Quadrupole: Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

41 Possible symmetries Lines of symmetry: Dipole Quad Pole orientation y = 0 x = 0 determines whether pole y = 0 is normal or skew. Additional symmetry x = 0 y = ± x imposed by pole edges. The additional constraints imposed by the symmetrical pole edges limits the values of n that have non-zero coefficients Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

42 Dipole symmetries Type Symmetry Constraint Pole orientation φ θ = φ( θ) all J n = 0 Pole edges φ(θ) = φ(π θ) K n non-zero only for n = 1, 3, 5, etc. +φ +φ +φ -φ So, for a fully symmetric dipole, only 6, 10, 14 etc. pole errors can be present. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

43 Quadrupole symmetries Type Symmetry Constraint Pole orientation φ(θ) = φ( θ) All J n = 0 φ(θ) = φ(π θ) K n = 0 for all odd n Pole edges φ(θ) = φ( ππ 2 θ) K n non-zero for n = 2, 6, 10, etc. So, for a fully symmetric quadrupole, only 12, 20, 28 etc. pole errors can be present. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

44 Sextupole symmetries Type Symmetry Constraint Pole orientation φ(θ) = φ( θ) All J n = 0 φ(θ) = φ( 2ππ θ) 3 K n = 0 where n is φ(θ) = φ( 4ππ θ) 3 not a multiple of 3 Pole edges φ(θ) = φ( ππ 3 θ) K n non-zero only for n = 3, 9, 15, etc. So, for a fully symmetric sextupole, only 18, 30, 42 etc. pole errors can be present. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

45 Summary - Allowed Harmonics Summary of allowed harmonics in fully symmetric magnets: Fundamental geometry Allowed harmonics Dipole, n = 1 n = 3, 5, 7,... (6 pole, 10 pole, etc.) Quadrupole, n = 2 n = 6, 10, 14,... (12 pole, 20 pole, etc.) Sextupole, n = 3 n = 9, 15, 21,... (18 pole, 30 pole, etc.) Octupole, n = 4 n = 12, 20, 28,... (24 pole, 40 pole, etc.) Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

46 Asymmetries generating harmonics (i) Two sources of asymmetry generate forbidden harmonics: i) magnetic asymmetries - significant at low permeability: e.g. C core dipole not completely symmetrical about pole centre, but negligible effect with high permeability. Generates n = 2,4,6, etc. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

47 Asymmetries generating harmonics (ii) ii) asymmetries due to small manufacturing errors in dipoles: n = 2, 4, 6 etc. n = 3, 6, 9, etc. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

48 Asymmetries generating harmonics (iii) ii) asymmetries due to small manufacturing errors in quadrupoles: n = 4 -ve n = 3 n = 2 (skew) n = 3 n = 4 +ve These errors are bigger than the finite µ type; can seriously affect machine behaviour and must be controlled. Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

49 Current Affairs FIELDS DUE TO COILS Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

50 Introduction of currents Now for jj 0 HH = jj To expand, use Stoke s Theorem for any vector V and a closed curve s: VV. ddss = VV. ddss Apply this to: HH = jj then in a magnetic circuit: (Ampere s equation) V ds ds HH. ddss = NNNN N I (Ampere-turns) is total current cutting S Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

51 Excitation current in a dipole B is approximately constant round the loop made up of λ and g, (but see below); But in iron, μμ 1 λ g NI/2 and So BB aaaaaa = μμ 0NNNN gg+ λλ μμ HH iiiiiiii = HH aaaaaa μμ µ >> 1 NI/2 g, and λ/µ are the 'reluctance' of the gap and iron. Approximation ignoring iron reluctance ( λλ μμ gg): NNNN = BBBB μμ 0 Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

52 Excitation current in quad & sextupole For quadrupoles and sextupoles, the required excitation can be calculated by considering fields and gap at large x. For example: Quadrupole: y Pole equation: xxxx = RR2 2 On x axes BB yy = gggg where g is gradient in T/m At large x (to give vertical lines of B): NNII = gggg RR2 2xxμμ 0 i.e. NNII = gggg2 (per pole) 2 μμ 0 x The same method for a Sextupole (coefficient g S ) gives: NNII = gg ssrr 3 (per pole) 3μμ 0 B Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

53 General solution for magnets order n In air (remote currents!) Integrating over a limited path BB = µ 0 HH BB = φ (not circular) in air: NNNN = φ 1 φ 2 μμ 0 φ 1, φ 2 are the scalar potentials at two points in air. Define φ = 0 at magnet centre; then potential at the pole is: µ 0 NNII y φ = µ 0 NI Apply the general equations for magnetic field harmonic order n for non-skew φ = 0 magnets (all J n = 0) giving: NNNN = 1 1 BB rr nn μμ 0 RRnn 1 RRnn Where: NI is excitation per pole R is the inscribed radius (or half gap in a dipole) RR BB rr nn 1 is magnet strength in T/m(n-1) Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn

54 Further Reading CERN Accelerator School on Magnets Bruges, Belgium; June United States Particle Accelerator School Magnet and RF Cavity Design, January J.D. Jackson, Classical Electrodynamics J.T. Tanabe, Iron Dominated Electromagnets Ben Shepherd, ASTeC Cockcroft Institute: Conventional Magnets, Autumn